MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide GEOMETRY HONORS Course Code:

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1 Topic II: Transformations in the Plane Pacing Date(s) Traditional 14 09/15/14-10/03/14 Block 07 09/15/14-10/03/14 MATHEMATICS FLORIDA STANDARDS & MATHEMATICAL PRACTICE (MP) MATHEMATICAL PRACTICE (MP) ESSENTIAL CONTENT OBJECTIVES MAFS.912.G-CO.1.2: Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). (MP.5, MP.6) Also Assesses: MAFS.912.G-CO.1.4: Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. (MP.5, MP.6, MP.7) MAFS.912.G-CO.1.5: Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. (MP.3, MP.5, MP.7) Also Assesses: MAFS.912.G-CO.1.3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. (MP.3, MP.5, MP.7) MAFS.K12.MP.1 Make sense of problems and persevere in solving them. MAFS.K12.MP.2 Reason abstractly and quantitatively. MAFS.K12.MP.3 Construct viable arguments and critique the reasoning of others. MAFS.K12.MP.4 Model with mathematics MAFS.K12.MP.5 Use appropriate tools strategically. MAFS.K12.MP.6 Attend to precision. MAFS.K12.MP.7 Look for and make use of structure. MAFS.K12.MP.8 Look for and express regularity in repeated reasoning. A. Rigid Transformations a) Reflection b) Rotations c) Translations d) Compositions of Transformations B. Non-Rigid Transformations C. Symmetry I can: Use definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments Represent transformations in the plane Draw transformations in the coordinate plane: reflections, rotations, translations, and combinations of these Complete transformations using a variety of tools: graph paper, transparencies, and/or geometry software Apply two or more transformations to a given figure to draw a transformed figure Identify the sequence of transformations (a composition) that will map a figure onto another Specify a sequence of transformations that will carry a figure onto another Describe transformations as functions Describe transformations as functions that take points in the plane as inputs and give other points as outputs Determine the coordinates for the image (output) of a figure when a transformation rule is applied to the pre-image (input). Compare transformations that preserve distance and angle to those that do not Describe rotations and reflections that carry a geometric figure onto itself Describe and illustrate how a rectangle, parallelogram, and isosceles trapezoid are mapped onto themselves using transformations. Calculate the number of lines of reflection symmetry and the degree of rotational symmetry of any regular polygon

2 INSTRUCTIONAL TOOLS Core Text Book: Discovering Geometry (4 th Edition) Pacing Date(s) Traditional 14 09/15/14-10/03/14 Block 07 09/15/14-10/03/14 Topic II Assessment Window 09/26/14-09/24/14 Standard Suggested Lessons Teacher Notes MAFS.912.G-CO.1.2 MAFS.912.G-CO.1.3 MAFS.912.G-CO.1.4 MAFS.912.G-CO Exploration page 595 Topic II Assessment: Transformations in the Plane EQuIP Rubric Dimensions The EQuIP Rubric Dimensions provide criteria to determine the quality and alignment of lessons and units to the MAFS in order to: (1) Identify exemplars/ models for teachers use within and across states; (2) provide constructive criteria-based feedback to developers; and (3) review existing instructional materials to determine what revisions are needed. Here are the four areas of your lesson to review: (For ratings information, go to EQuIP Rubric) I. Alignment to the Depth of the MAFS The lesson/unit aligns with the letter and spirit of the MAFS. II. Key Shifts in the MAFS The lesson/unit reflects evidence of key shifts that are reflected in MAFS: Focus, Coherence, and Rigor III. Instructional Supports The lesson/unit is responsive to varied student learning needs. IV. Assessment The lesson/unit regularly assesses whether students are mastering standards-based content and skills. Algebra 1 (NGSSS) Opportunities for Remediation Algebra Nation (NGSSS) Section 1: Sets and Venn Diagrams Section 2: Relations and Functions Section 3: Solving for X Section 4: Solving Real World Equations Housed in: Learning Village/ Pacing Guides and Curriculum/ High School/ Mathematics/ Algebra 1 (NGSSS) Opportunities for Remediation/ Topic II MiniPractice MA.912.D.7.1 MiniPractice MA.912.D.7.2 MiniPractice MA.912.A.2.3 MiniPractice MA.912.A.2.4 MiniPractice MA.912.A.3.1 MiniPractice MA.912.A.3.3 & A.5.4 MiniPractice MA.912.A.3.4 MiniPractice MA.912.A.3.5

3 INSTRUCTIONAL TOOLS Formative Assessments Name Demonstrating Rotations: Demonstrating Reflections: Transformations And Functions: Comparing Transformations: Demonstrating Translations: Transformations of Trapezoids: Transformations of Regular Polygons: Transformations of Rectangles and Squares: Transformations of Parallelograms and Rhombi: Define a Rotation: Define a Reflection: Define a Translation: Two Triangles: Reflect a Semicircle: Indicate the Transformations: Rotation of a Quadrilateral: Description Students are asked to rotate a quadrilateral 90 degrees clockwise. Students are asked to reflect a quadrilateral across a given line. Students are given examples of three transformations and are asked if each is a function. Students are asked to determine whether or not dilations and reflections preserve distance and angle measure. Students are asked to translate a quadrilateral according to a given vector. Students are asked to describe the rotations and reflections that carry a trapezoid onto itself. Students are asked to describe the rotations and reflections that carry a regular polygon onto itself. Students are asked to describe the rotations and reflections that carry a rectangle and a square onto itself. Students are asked to describe the rotations and reflections that carry a parallelogram and rhombus onto itself. Students are asked to develop a definition of rotation. Students are asked to develop a definition of reflection. Students are asked to develop a definition for translation. Students are asked to describe the transformations that take one triangle onto another. Students draw the reflection of a given semicircle across a reflection line. Students are asked to describe the transformations that take one triangle onto another. Students are asked to rotate a quadrilateral around a given point. Vocabulary: Image, Transformation, Rigid Transformation, Isometry, Non-rigid Transformation, Translation, Translation Vector, Reflectional Symmetry, Line of Symmetry, Rotational Symmetry, Point Symmetry, Ordered Pair Rules, Minimal Path Instructional Strategies: Review vocabulary associated with transformations (e.g. point, line, segment, angle, circle, polygon, parallelogram, perpendicular, rotation reflection, translation). Provide both individual and small-group activities, allowing adequate time for students to explore and verify conjectures about transformations and develop precise definitions of rotations, reflections and translations. Provide real-world examples of rigid motions (e.g. Ferris wheels for rotation; mirrors for reflection; moving vehicles for translation). Use graph paper, transparencies, tracing paper or dynamic geometry software to obtain images of a given figure under specified transformations. Provide students with a pre-image and a final, transformed image, and ask them to describe the steps required to generate the final image. Show examples with more than one answer (e.g., a reflection might result in the same image as a translation).

4 INSTRUCTIONAL TOOLS Work backwards to determine a sequence of transformations that will carry (map) one figure onto another of the same size and shape. Focus attention on the attributes (e.g. distances or angle measures) of a geometric figure that remain constant under various transformations. Make the transition from transformations as physical motions to functions that take points in the plane as inputs and give other points as outputs. The correspondence between the initial and final points determines the transformation. Analyze various figures (e.g. regular polygons, folk art designs or product logos) to determine which rotations and reflections carry (map) the figure onto itself. These transformations are the symmetries of the figure. Emphasize the importance of understanding a transformation as the correspondence between initial and final points, rather than the physical motion. Remarks: Students should know that not every transformation is a translation. Students sometimes confuse the terms transformation and translation. Students may confuse rotations and reflections and be unable to differentiate the two. Allowing them the opportunity to physically manipulate the shapes (such as with cut-outs or patty paper) can clear up misconceptions. Remind students that that corresponding vertices have to be listed in order so that corresponding sides and angles can be easily identified. Connections: Rotations, reflections and translations are developed experimentally in the Grade 8 (MAFS), and this experience should be built upon in high school, giving greater attention to precise definitions and formal reasoning.

5 MATHEMATICS FLORIDA STANDARDS MATHEMATICAL PRACTICES DESCRIPTION MAFS.K12.MP.1 Make sense of problems and persevere in solving them. Explain the meaning of a problem and looking for entry points to its solution. Analyze givens, constraints, relationships, and goals. Make conjectures about the form and meaning of the solution and plan a solution pathway. Consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. Monitor and evaluate their progress and change course if necessary. Explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Check answers to problems using a different method, and continually ask, Does this make sense? Identify correspondences between different approaches. MAFS.K12.MP.2 Reason abstractly and quantitatively. Make sense of quantities and their relationships in problem situations. Decontextualize to abstract a given situation and represent it symbolically. Contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols Create a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them. Know and be flexible using different properties of operations and objects. MAFS.K12.MP.3 Construct viable arguments and critique the reasoning of others. Understand and use stated assumptions, definitions, and previously established results in constructing arguments. Make conjectures and build a logical progression of statements to explore the truth of their conjectures. Analyze situations by breaking them into cases, and can recognize and use counterexamples. Justify their conclusions, communicate them to others, and respond to the arguments of others. Reason inductively about data, making plausible arguments that take into account the context from which the data arose. Compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument explain what it is. Determine domains to which an argument applies. MAFS.K12.MP.4 Model with mathematics. Apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. Use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Apply what they know and feel comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. Identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. Analyze relationships mathematically to draw conclusions. Interpret mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

6 MATHEMATICS FLORIDA STANDARDS MATHEMATICAL PRACTICES DESCRIPTION MAFS.K12.MP.5 Use appropriate tools strategically. Consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Make sound decisions about when each of the tools appropriate for their grade or course might be helpful, recognizing both the insight to be gained and their limitations. Example: High school students analyze graphs of functions and solutions using a graphing calculator. Detect possible errors by strategically using estimation and other mathematical knowledge. Know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. Use technological tools to explore and deepen their understanding of concepts MAFS.K12.MP.6 Attend to precision. Communicate precisely to others. Use clear definitions in discussion with others and in their own reasoning. State the meaning of the symbols they choose, including using the equal sign consistently and appropriately. Be careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. Calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. MAFS.K12.MP.7 Look for and make use of structure. Discern a pattern or structure. Example: In the expression x 2 + 9x + 14, students can see the 14 as 2 7 and the 9 as Recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. Step back for an overview and shift perspective. See complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. Example: They can see 5 3(x y) 2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. MAFS.K12.MP.8 Look for and express regularity in repeated reasoning. Notice if calculations are repeated, and look both for general methods and for shortcuts. Example: Noticing the regularity in the way terms cancel when expanding (x- 1)(x+1),(x-1)(x2+x+1),and(x-1)(x3 +x2+x+1)might lead them to the general formula for the sum of a geometric series. Maintain oversight of the process, while attending to the details as they work to solve a problem. Continually evaluate the reasonableness of their intermediate results.

7 Domain: GEOMETRY: CONGRUENCE STANDARD CODE Cluster 1: Experiment with transformations in the plane MAFS.912.G-CO.1.2 Assessed MATHEMATICS FLORIDA STANDARDS STANDARD DESCRIPTION Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). Context Complexity: Level 2: Basic Application of Skills & Concepts MAFS.912.G-CO.1.3 (Assessed with MAFS.912.G-CO.1.5) MAFS.912.G-CO.1.4 (Assessed with MAFS.912.G-CO.1.5) MAFS.912.G-CO.1.5 Assessed Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. Context Complexity: Level 2: Basic Application of Skills & Concepts Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. Context Complexity: Level 3: Strategic Thinking & Complex Reasoning Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Context Complexity: Level 2: Basic Application of Skills & Concepts

8 TECHNOLOGY TOOLS CPALM RESOURCES LESSON PLANS Fundamental Property of Reflections Transform through the Maze Rotation of Polygons about a Point: Reflections Hands On VIRTUAL MANIPULATIVE Transformations - Translation Transformations - Rotation Playing with Reflections PROBLEM-SOLVING TASK Seven Circles II Reflected Triangles GIZMO CORRELATION GIZMO TITLE Rock Art Reflections Translations Rotations, Reflections and Translations GEOMETER S SKETCHPAD ACTIVITIES UNIT/ TITLE Rotation & Translation Exploration Transformations TOPIC II VIDEO TITLE IMAGE

9 TOPIC II VIDEO TITLE DISCOVERY EDUCATION CORRELATION Example 2: Rotation and Translation -- The London Eye and Boating Rotational Symmetry Symmetry and Transformations MATH EXPLANATION TITLE Geometry: Translations and Glide Reflections: Graphing Translations of Segments Geometry: Reflections: Translations and Reflections Geometry: Translations and Glide Reflections: Graphing Translations of Figures Geometry: Translations and Glide Reflections: Reflections and Translations Geometry: Reflections: Drawing Line Reflections Geometry: Reflections: Reflection When the Figure Is Graphed Geometry: Reflections: Double Reflections, Part One Geometry: Reflections: Real - World Reflections Geometry: Reflections: Coordinate Plane Double Reflections Geometry: Reflections: Double Reflections, Part Two MATH OVERVIEW Geometry: Translations and Glide Reflections STUDENT S e-resources KHAN ACADEMY Transformations Symmetry Exploring Rigid Transformations Transformations on the Coordinate Plane Let's get an intuitive understanding for symmetry of two dimensional shapes Let's use some pretty cool software widgets to see how we can transform a shape through translations, rotations, dilations and reflections. Let's continue our deep voyage through the world of transformations by thinking about how points map to each other after a transformation.

10 Date Traditional 14 days Block 7 days Pacing Guide Standards MAFS.912.G-CO.1.2 MAFS.912.G-CO.1.3 MAFS.912.G-CO.1.4 MAFS.912.G-CO.1.5 Data Driven Standard(s) Activities Assessment(s) Strategies 09/15/14-10/03/14